Question

A coil of $N$ turns and radius $R$ carries a current $I$. It is unwound and rewound to make a square coil of side $a$ having same number of turns $N$. Keeping the current $I$ same find the ratio of the magnetic moments of the square coil and the circular coil.

Answer 1

Magnetic moment of current loop,$M=NIA$

For circular loop $Mc=NI\pi R^2$

When the coil is unwound and rewound to make a square coil, then 2∏R =4a

$a=\frac{\pi R}{2}$

Hence, Magnetic moment of square coil $Ms=NI{a}^2$

$=NI{\left[\frac{\left(\pi R\right)}{2}\right]}^2$

$=\frac{\left(NI{\pi }^2{R}^2\right)}{4}$

Now, ratio of magnetic moments of the square coil and circular coil

$Ms/Mc=\frac{\frac{\left(NI{\pi }^2{R}^2\right)}{4}}{NI\pi R^2}$

$=\frac{\pi }{4}$